This is a random question that has been bugging me. In first-year calculus we learned that the second-order linear homogeneous ODE with complex roots $a\pm ib$ to its characteristic equation , has a real-valued general solution of the form: $y(x)=e^{ax}\left(c_1cos(bx)+c_2sin(bx)\right)$.
To get to this real-valued general solution, some intermediate steps were performed on the original complex-valued general solution, $y(x)=Pe^{(a+ib)x}+Qe^{(a-ib)x}$.
I'm interested in learning why the real-valued case is of particular interest only. Is there a particular reason for doing so?
Best Answer
If the initial conditions $y(0)$ and $y'(0)$ are real numbers, then the solution $y(x)$ must be real-valued. And this situation is what you have in most applications.